Mixture and hidden Markov models with R is written by Ingmar Visser and Maarten Speekenbrink.
The book aims to provide a self-contained practical introduction to mixture models and hidden Markov models. The reason for introducing both in one book is that there are very close links between these models. This allows us to introduce important concepts, such as maximum likelihood estimation and the Expectation-Maximization algorithm, in the relatively simpler context of mixture models. Approaching hidden Markov models from a thorough understanding of mixture models involves, we hope, a relatively small conceptual leap.
We aimed to provide a reasonable balance between statistical theory and practice. The objective is to provide enough mathematical details to allow our target audience to understand key results that are necessary to apply these models. Our target audience are those with a more applied background, in particular researchers, graduate, and advanced undergraduate students in the social and behavioural sciences. Researchers or future researchers hence who see the potential for applying these models and explaining heterogeneity in their data, but who lack the tools to fulfil this potential.
To familiarise readers with the possibilities of mixture and hidden Markov models,
a large part of the book consists of practical examples of applying
these models, many of which taken from our own research in
developmental and experimental psychology, as well as from other fields,
such as climate change and economics. These examples show how to analyse
mixture and hidden Markov models with R, with a particular focus on our
depmixS4 package, as well as the accompanying hmmr package which contains
all the datasets used in the book, as well as a number of additional convenience
functions.
The book consists of seven chapters:
Chapter 1 provides a brief introduction to R, and describes the datasets analysed in the examples.
Chapter 2 is a theoretical chapter on mixture models. It covers the definition of these models, methods for maximum likelihood parameter estimation, parameter inference via likelihood ratio tests and confidence intervals, model selection techniques to e.g. determine the number of mixture components, how to model the effect of covariates on the prior probability of the components, and identifiability of mixture models.
Chapter 3 is an applied chapter with several applications of mixture and latent class models. These include of univariate and multivariate Gaussian mixture models for financial and psychological data, a latent class model for multivariate binary data, and binomial mixture models. Topics treated with these applications include local maxima in the likelihood and other practical difficulties in model estimation, item homogeneity, direct effects of covariates on responses vs indirect effects on the prior probability of mixture components, and use of the bootstrapped likelihood ratio test to determine the number of mixture components.
Chapter 4 is another theoretical chapter, focusing on hidden Markov models. Building on Chapter 2, this chapter shows how hidden Markov models can be viewed as an extension of mixture models to allow dependency between the states (mixture components) at consecutive time points. Important properties of Markov chains such as stationarity, homogeneity, and ergodicity, are discussed. After moving on to hidden Markov models, inference of the hidden states via filtering and smoothing recursions are treated in detail, as well as state decoding via the Viterbi algorithm. Other topics include the use of covariates to model initial state and state transition probabilities, and dealing with missing data.
Chapter 5 is an applied chapter which describes applications of hidden Markov models (HMMs) for univariate time series. These include a Gaussian HMM for financial time series, a Bernoulli HMM applied to a relatively large number of relatively short timeseries, a Gaussian HMM applied to (log) response times, a Gaussian HMM to detect change points in climate change data, and HMMs with generalized linear models, which can be viewed as modelling a “switching GLM”. Some topics treated with these examples include accounting for autocorrelation in timeseries, constraining state transitions to detect change-points, and using different covariates to predict responses in different states.
Chapter 6 is another applied chapter which focuses on hidden Markov models for multivariate time series. These include a multivariate Binomial HMM for a large number of replications of relative short timeseries, showcasing how HMMs can be used to analyse complex panel data. Other application concerns a HMMs for mixed data, consisting of a Bernoulli and a Gaussian variable, and consisting of a binomial and multinomial variable. Some of the topics treated in these applications include testing hysteresis, testing conditional independence of responses within states, and models in which one response variable can act as a predictor for another response variable.
Chapter 7 is a mostly theoretical chapter, discussing several extensions of the material presented thus far. This includes expanding the state-space to allow for higher-order Markov models (where the current state can depend on states further in the past than the immediately preceding state) and models with multiple simultaneous hidden states. The “classification likelihood” is discussed as an alternative to the usual likelihood function, and techniques for dealing with practical estimation issues are also described, as well as using Bayesian estimation and inference.
Ingmar Visser is an Associate Professor in Developmental Psychology at the University of Amsterdam. His core interest is in characterizing the building blocks of the human cognitive architecture, such as the concepts we use to classify objects in the world around us. These building blocks are shaped during learning and development, for example in category learning and in implicit learning. Across different experimental paradigms he develops state-of-the-art analytical approaches to further understanding of their results. Obviously, these methods include mixture and hidden Markov models.
Maarten Speekenbrink is an Associate Professor in Mathematical Psychology at University College London. His research focuses mostly on human learning and decision making, adopting the formal framework of reinforcement learning to address topics such as how uncertainty drives exploration and exploitation, and how useful representations are formed and learned to support further learning and generalization. He develops and applies advanced statistical methods to investigate these topics, including Gaussian Process regression, particle filters, and of course mixture and hidden Markov models. He teaches advanced statistics to postgraduate students, and has written several tutorials on the aforementioned techniques.
The authors have known each other since their days as PhD students at the department of Psychology at the University of Amsterdam, and have remained friends ever since. Large parts of the book were written during intensive stays in Amsterdam, London, and Berlin, dispersed by long hiatuses where other priorities took over.